Question bank fourier series Engg. Mathematics- ii

JAHANGIRABAD INSTITUTEOF

TECHNOLOGY

FOURER SERIES (QUESTION BANK)

ENGINEERING MATHEMATICS –II

PREPRAED BYMOHAMMAD IMRAN

(ASSISTANT PROFESSOR, JIT)

E-mail: [email protected]

Website: www.jit.edu.in

http://www.jit.edu.in/

Mobile no 9648588546

FOURER SERIESDEFINITION :

An infinite Series of Trigonometric functions which represents an expansion or approximation of a periodic function, used in Fourier analysis.

EVEN FUNCTION:

Let f(x) be a real-valued function of a real variable.

Then f is even if the following equation holds for all

x and -x in the domain of f. or Geometrically speaking, the graph face of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.Examples of even functions are |x|, x2, x4, cos(x), cosh(x), or any linear combination of these.

ODD FUNCTIONS

Again, let f(x) be a real-valued function of a real variable.

Then f is odd if the following equation holds for all

x and -x in the domain of f:

or

Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin.

Examples of odd functions are x, x3, sin(x), sinh(x), erf(x), or

any linear combination of these.

Properties involving addition and subtraction

The sum of two even functions is even, and any constant multiple of an even function is even. The sum of two odd functions is odd, and any constant multiple of an odd function is odd. The difference between two odd functions is odd. The difference between two even functions is even. The sum of an even and odd function is neither even nor odd, unless one of the functions is equal to zero over

the given domain.

Properties involving multiplication and division

The product of two even functions is an even function. The product of two odd functions is an even function. The product of an even function and an odd function is an odd function. The quotient of two even functions is an even function. The quotient of two odd functions is an even function. The quotient of an even function and an odd function is an odd function.

Properties involving composition

The composition of two even functions is even. The composition of two odd functions is odd. The composition of an even function and an odd function is even. The composition of any function with an even function is even (but not vice versa).

Other algebraic properties

Any linear combination of even functions is even, and the even functions form a vector space over the reals. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. In fact, the vector space of all real-valued functions is the direct sum of the subspaces of even and

odd functions. In other words, every function f(x) can be written uniquely as the sum of an even function and an odd function:

where

is even and

is odd. For example, if f is exp, then fe is cosh and fo is sinh.

The even functions form a commutative algebra over the reals. However, the odd functions do not form an algebra over the reals, as they are not closed under multiplication.

IMPORTANT QUESTIONS

1. find the Fourier series for the function uptu 2009



hence show that

4. find the Fourier series for the function uptu 2002, 2008

Hence show that



Deduce that


8. find the Fourier series for the function


Deduce that


11. Obtain the half range sine series for the function in the interval uptu 2008

12. Find the half range cosine series for the function uptu 2001,06, 0713. Obtain the fourier cosine series expansion of the periodic function define by the function uptu 2001

14. Find the half range sine series for the function given in the range by the graph OPQ as shown in the figure. Uptu 2009

Engineering

Question bank fourier series Engg. Mathematics- ii